# Problem of the Day: 1/30/13

If $2^{2n+3}=\frac{2^{n-2}}{2^{2n-2}}$, what is n?

(This equation has been corrected from yesterday.)

Solution to yesterday’s problem:

Here’s the diagram again, with some small additions:

Notice that each of the four triangles is a 30-60-90. This means we can find the side lengths of all the sides given one of the sides. Recall that the short side is n, the hypotenuse is 2n, and the long side is n times the square root of 2. We know that 2n = 8 cm, so the short side must be 4cm. Now we can make another interesting addition to the diagram considering that a tangent line is perpendicular to a circle’s radius:

Now we have another 30-60-90 triangle because one of the angles is the same, and one of the angles is 90°. Knowing that the hypotenuse is 4 cm, the short side is 2 cm, and the long side is $2\sqrt{3}$ cm.